Optimal. Leaf size=573 \[ \frac {(b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right )}{10 c^{5/2} \sqrt {b^2-4 a c} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}+\frac {\left (b^2-4 a c\right )^{3/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{20 \sqrt {2} c^{11/4} (b+2 c x)}-\frac {\left (b^2-4 a c\right )^{3/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{10 \sqrt {2} c^{11/4} (b+2 c x)}+\frac {7 e \left (a+b x+c x^2\right )^{3/4} (2 c d-b e)}{15 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{3/4}}{5 c} \]
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Rubi [A] time = 0.73, antiderivative size = 573, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {742, 640, 623, 305, 220, 1196} \[ \frac {(b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right )}{10 c^{5/2} \sqrt {b^2-4 a c} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}+\frac {\left (b^2-4 a c\right )^{3/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{20 \sqrt {2} c^{11/4} (b+2 c x)}-\frac {\left (b^2-4 a c\right )^{3/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{10 \sqrt {2} c^{11/4} (b+2 c x)}+\frac {7 e \left (a+b x+c x^2\right )^{3/4} (2 c d-b e)}{15 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{3/4}}{5 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 623
Rule 640
Rule 742
Rule 1196
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\sqrt [4]{a+b x+c x^2}} \, dx &=\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{3/4}}{5 c}+\frac {2 \int \frac {\frac {1}{4} \left (10 c d^2-4 e \left (\frac {3 b d}{4}+a e\right )\right )+\frac {7}{4} e (2 c d-b e) x}{\sqrt [4]{a+b x+c x^2}} \, dx}{5 c}\\ &=\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{15 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{3/4}}{5 c}+\frac {\left (-\frac {7}{4} b e (2 c d-b e)+\frac {1}{2} c \left (10 c d^2-4 e \left (\frac {3 b d}{4}+a e\right )\right )\right ) \int \frac {1}{\sqrt [4]{a+b x+c x^2}} \, dx}{5 c^2}\\ &=\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{15 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{3/4}}{5 c}+\frac {\left (4 \left (-\frac {7}{4} b e (2 c d-b e)+\frac {1}{2} c \left (10 c d^2-4 e \left (\frac {3 b d}{4}+a e\right )\right )\right ) \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{5 c^2 (b+2 c x)}\\ &=\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{15 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{3/4}}{5 c}+\frac {\left (2 \sqrt {b^2-4 a c} \left (-\frac {7}{4} b e (2 c d-b e)+\frac {1}{2} c \left (10 c d^2-4 e \left (\frac {3 b d}{4}+a e\right )\right )\right ) \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{5 c^{5/2} (b+2 c x)}-\frac {\left (2 \sqrt {b^2-4 a c} \left (-\frac {7}{4} b e (2 c d-b e)+\frac {1}{2} c \left (10 c d^2-4 e \left (\frac {3 b d}{4}+a e\right )\right )\right ) \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {2 \sqrt {c} x^2}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{5 c^{5/2} (b+2 c x)}\\ &=\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{15 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{3/4}}{5 c}+\frac {\left (20 c^2 d^2+7 b^2 e^2-4 c e (5 b d+2 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{10 c^{5/2} \sqrt {b^2-4 a c} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}-\frac {\left (b^2-4 a c\right )^{3/4} \left (20 c^2 d^2+7 b^2 e^2-4 c e (5 b d+2 a e)\right ) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{10 \sqrt {2} c^{11/4} (b+2 c x)}+\frac {\left (b^2-4 a c\right )^{3/4} \left (20 c^2 d^2+7 b^2 e^2-4 c e (5 b d+2 a e)\right ) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{20 \sqrt {2} c^{11/4} (b+2 c x)}\\ \end {align*}
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Mathematica [C] time = 0.21, size = 165, normalized size = 0.29 \[ \frac {\frac {3 (b+2 c x) \sqrt [4]{\frac {c (a+x (b+c x))}{4 a c-b^2}} \left (-4 c e (2 a e+5 b d)+7 b^2 e^2+20 c^2 d^2\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{2 \sqrt {2} c^2}-\frac {14 e (a+x (b+c x)) (b e-2 c d)}{c}+12 e (d+e x) (a+x (b+c x))}{30 c \sqrt [4]{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{2} x^{2} + 2 \, d e x + d^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^2}{{\left (c\,x^2+b\,x+a\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{2}}{\sqrt [4]{a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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